Optimal. Leaf size=92 \[ 2 i \sin ^{-1}(a x) \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-2 i \sin ^{-1}(a x) \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-2 \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+2 \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )-2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.14, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4709, 4183, 2531, 2282, 6589} \[ 2 i \sin ^{-1}(a x) \text {PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-2 i \sin ^{-1}(a x) \text {PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-2 \text {PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+2 \text {PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )-2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4183
Rule 4709
Rule 6589
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx &=\operatorname {Subst}\left (\int x^2 \csc (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-2 \operatorname {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+2 \operatorname {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+2 i \sin ^{-1}(a x) \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-2 i \sin ^{-1}(a x) \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-2 i \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+2 i \operatorname {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+2 i \sin ^{-1}(a x) \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-2 i \sin ^{-1}(a x) \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )+2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )\\ &=-2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+2 i \sin ^{-1}(a x) \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-2 i \sin ^{-1}(a x) \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-2 \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+2 \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.13, size = 116, normalized size = 1.26 \[ 2 i \sin ^{-1}(a x) \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-2 i \sin ^{-1}(a x) \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-2 \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+2 \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )+\sin ^{-1}(a x)^2 \log \left (1-e^{i \sin ^{-1}(a x)}\right )-\sin ^{-1}(a x)^2 \log \left (1+e^{i \sin ^{-1}(a x)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a^{2} x^{3} - x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arcsin \left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 161, normalized size = 1.75 \[ \arcsin \left (a x \right )^{2} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 i \arcsin \left (a x \right ) \polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+2 \polylog \left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )-\arcsin \left (a x \right )^{2} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+2 i \arcsin \left (a x \right ) \polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 \polylog \left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arcsin \left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{x\,\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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